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Abstract

Spatial adiabatic passage represents a new way to design integrated photonic devices. In conventional adiabatic passage, designs require smoothly varying waveguide separations. Here we show modelling of adiabatic passage devices where the waveguide separation is varied digitally. Despite digitisation, our designs show robustness against variations in the input wavelength and refractive index contrast of the waveguides relative to the cladding. This approach to spatial adiabatic passage opens new design strategies and hence the potential for new photonics devices.

E. A. Shapiro, V. Milner, and M. Shapiro, “Complete transfer of populations from a single state to a preselected superposition of states using piecewise adiabatic passage: Theory,” Phys. Rev. A 79, 023422 (2009).
[Crossref]

E. A. Shapiro, V. Milner, and M. Shapiro, “Complete transfer of populations from a single state to a preselected superposition of states using piecewise adiabatic passage: Theory,” Phys. Rev. A 79, 023422 (2009).
[Crossref]

Menzel-Jones, C.

Milner, V.

E. A. Shapiro, V. Milner, and M. Shapiro, “Complete transfer of populations from a single state to a preselected superposition of states using piecewise adiabatic passage: Theory,” Phys. Rev. A 79, 023422 (2009).
[Crossref]

Rangelov, A.

Rangelov, A. A.

A. A. Rangelov and N. V. Vitanov, “Complete population transfer in a three-state quantum system by a train of pairs of coincident pulses,” Phys. Rev. A 85, 043407 (2012).
[Crossref]

Shapiro, E. A.

E. A. Shapiro, V. Milner, and M. Shapiro, “Complete transfer of populations from a single state to a preselected superposition of states using piecewise adiabatic passage: Theory,” Phys. Rev. A 79, 023422 (2009).
[Crossref]

Shapiro, M.

E. A. Shapiro, V. Milner, and M. Shapiro, “Complete transfer of populations from a single state to a preselected superposition of states using piecewise adiabatic passage: Theory,” Phys. Rev. A 79, 023422 (2009).
[Crossref]

E. A. Shapiro, V. Milner, and M. Shapiro, “Complete transfer of populations from a single state to a preselected superposition of states using piecewise adiabatic passage: Theory,” Phys. Rev. A 79, 023422 (2009).
[Crossref]

Figures (8)

(a) Structure for digital waveguide adiabatic passage showing the segmented waveguide with circular geometry. The counter-intuitive coupling sequence is achieved by light propagating in the z-direction entering at the bottom left waveguide, and exiting via the top right, with the coupling mediated in the x-direction by the central waveguidelets (shown colored). Figures (b), (c) and (d) show the refractive index profiles for the red, green and blue cases from (a), demonstrating the additive nature of a continuous refractive index profile. The red lines are the refractive index of each element independently, and the blue lines show the sum of the refractive indices. When the central waveguidelet is closest to one of the outer waveguides, the independent waveguide approximation breaks down. The last two waveguidelet images are mirror images of the first two and are not shown for brevity. Note that these images are purely for illustrative purposes only; the particular device parameters can be found in Table 1.

(a) Effective change to propagation constant due to the presence of another guide. (b) Numerically (solid) and semi-analytically (dashed) obtained coupling of Gaussian index fibers. The minimum separation is 2ρ so that the waveguides are clearly distinguishable. Device parameters are given in Table 1.

(a) Pseudo-colour plot showing the final state population (colour axis) as a function of δ and λ.(b) Pseudo-colour plot showing final state population as a function of λ and device length, L. In both cases note the wide wavelength range over which devices provide high-fidelity transport. The fidelity is periodic, and we have highlighted only one period here. The dark patch in the top right of the length subfigure is actually a pessimal resonance [12] with 90% in the initial state despite being designed for a completely different length and wavelength. All other parameters not being varied are the same as they are in Table 1.

Pb = |〈E2|b〉|2 as a function of scaled perturbative parameters using analytical (solid) and approximate (dashed) forms for Ωac (left) and βdiff (right). Values are symmetric with respect to Ωab ↔ Ωbc, to represent data in reduced units we divide through all parameters by Ωab. We also divide the perturbative parameters by Ωbc, as any value exceeding Ωbc/2 would no longer be a perturbation. The functions each have a local maximum at
Ωbc=Ωab/5 and
Ωbc=Ωab/2 (see (17) and (19) respectively) and so values are linearly spaced up to those points. These data show that the approximations are good over a wide range of possible values with deviations strongest at the turning point.

Illustrations of the mirrored five-state device considered in this subsection. (left) schematic demonstrating the couplings of the device and (right) illustration demonstrating the mirror geometry, each color corresponds to a set of waveguides described in Table 2. Light is injected in |c〉 and coupled into a superposition of |a〉 and |e〉. For simplicity, we will only consider when Ωbc = Ωcd and Ωab = Ωde resulting in a equal division of power. The x-coordinate describes the centre of the waveguide. Segments are varied in the x-direction to vary the couplings. All waveguides are placed such that no next-nearest neighbour coupling is possible.

Populations Pi = |〈ψ|i〉|2 during the five-state digital adiabatic 50:50 power division protocol on a regular (left) and log (right) scale. Faint dashed lines show the end/beginning of waveguidelets. It can be seen that the population traces Pa, Pe and Pb, Pd are directly on top of each other and that population in the intermediate waveguides goes to zero at the end of each waveguidelet. Coupling values were chosen to transfer equal population per step and device parameters can be found in Table 2.

(left) Illustration demonstrating couplings in the triangular four-state system considered in this subsection. (right) Schematic to-scale side-view of the parameters found in Table 3, waveguide sizes are decreased for distinguishability and the inset provided demonstrates the relatively small vertical movement. Light is injected into |a〉 and is brought into a controlled superposition of |c〉 and |d〉 by controlling the ratio
α=Ωbc/Ωbd=Pc/Pd. The x-coordinate describes the centre of the waveguide. The x and y-coordinates describe the centre of the waveguides. Segments are varied in the x and y-directions to vary the couplings while maintaining a constant ratio of couplings
α=Ωbc/Ωbd=Pc/Pd. All waveguides are placed such that no next-nearest neighbour coupling is possible.

Populations Pi = |〈ψ|i〉|2 during the four-state adiabatic 1/3:2/3 power division protocol on a regular (left) and log (right) scale. Faint dashed lines show the end/beginning of waveguidelets. It can be seen that the population in the intermediate waveguides goes to zero at the end of each waveguidelet. Coupling values were chosen to transfer equal population per step and device parameters can be found in Table 3.

Tables (3)

Table 1 Device geometry and parameters used in all calculations regarding the three-state coupler. DAP is from |a〉 to |c〉, and the central waveguide, |b〉, is split into 5 waveguidelets, |b〉1 to |b〉5. Propagation occurs in the z-direction and all segments are aligned at y = 0. |a〉, |b〉1, |c〉 all begin at z = 0. Segment |b〉i+1 is connected to the end of segment |b〉i. The x-coordinate describes the centre of the waveguide. Segments are varied in the x-direction to vary the couplings.

Table 2 Device geometry and parameters used in transport calculations for five-state based 50:50 splitter. DAP is from |c〉 to |a〉 and |e〉, and the intermediate waveguides, |b〉, |d〉, are split into 5 waveguidelets each, |b〉1, |d〉1 to |b〉5, |d〉5. Propagation occurs in the z-direction and all segments are aligned at y = 0. |a〉, |b〉1, |c〉, |d〉1, |e〉 all begin at z = 0. Segment |b〉i+1 (|d〉i+1) is connected to the end of segment |b〉i (|d〉i). The x-coordinate describes the centre of the waveguide. Segments are varied in the x-direction to vary the couplings. Recall that the device is symmetric about |c〉, and the positions of |a〉, |e〉 and |b〉, |d〉 are related by x → −x. Properties ρ, δ, λopt are the same as Table 1.

Table 3 Device geometry and parameters used in transport calculations for four-state based 1/3:2/3 splitter. DAP is from |a〉 to |c〉 and |d〉, and the intermediate waveguide, |b〉, is split into 5 waveguidelets, |b〉1 to |b〉5. Propagation occurs in the z-direction. |a〉, |b〉1, |c〉, |d〉 all begin at z = 0. Segment |b〉i+1 is connected to the end of segment |b〉i. The x and y-coordinates describe the centre of the waveguides. Segments are varied in the x and y-directions to vary the couplings while maintaining a constant ratio of couplings
Ωbc/Ωbd=Pc/Pd. Properties ρ, δ, λopt are the same as Table 1. Distances between |a〉, |c〉, and |d〉 are all 21μm to minimize cross-talk.

Metrics

Table 1

Device geometry and parameters used in all calculations regarding the three-state coupler. DAP is from |a〉 to |c〉, and the central waveguide, |b〉, is split into 5 waveguidelets, |b〉1 to |b〉5. Propagation occurs in the z-direction and all segments are aligned at y = 0. |a〉, |b〉1, |c〉 all begin at z = 0. Segment |b〉i+1 is connected to the end of segment |b〉i. The x-coordinate describes the centre of the waveguide. Segments are varied in the x-direction to vary the couplings.

Waveguidelet

|a〉

|b〉1

|b〉2

|b〉3

|b〉4

|b〉5

|c〉

Lopt(mm)

N/A

7.869

11.270

11.804

11.270

7.869

N/A

x (μm)

10.500

−1.177

−0.355

0.000

0.355

1.177

−10.500

ρ

3 μm

δ

0.0045

λopt

800 nm

Table 2

Device geometry and parameters used in transport calculations for five-state based 50:50 splitter. DAP is from |c〉 to |a〉 and |e〉, and the intermediate waveguides, |b〉, |d〉, are split into 5 waveguidelets each, |b〉1, |d〉1 to |b〉5, |d〉5. Propagation occurs in the z-direction and all segments are aligned at y = 0. |a〉, |b〉1, |c〉, |d〉1, |e〉 all begin at z = 0. Segment |b〉i+1 (|d〉i+1) is connected to the end of segment |b〉i (|d〉i). The x-coordinate describes the centre of the waveguide. Segments are varied in the x-direction to vary the couplings. Recall that the device is symmetric about |c〉, and the positions of |a〉, |e〉 and |b〉, |d〉 are related by x → −x. Properties ρ, δ, λopt are the same as Table 1.

Waveguidelet

|a〉

|b〉1

|b〉2

|b〉3

|b〉4

|b〉5

|c〉

Lopt(mm)

N/A

6.606

9.471

9.924

9.480

6.626

N/A

x (μm)

−21.000

−11.959

−11.138

−10.783

−10.427

−9.606

0.000

Table 3

Device geometry and parameters used in transport calculations for four-state based 1/3:2/3 splitter. DAP is from |a〉 to |c〉 and |d〉, and the intermediate waveguide, |b〉, is split into 5 waveguidelets, |b〉1 to |b〉5. Propagation occurs in the z-direction. |a〉, |b〉1, |c〉, |d〉 all begin at z = 0. Segment |b〉i+1 is connected to the end of segment |b〉i. The x and y-coordinates describe the centre of the waveguides. Segments are varied in the x and y-directions to vary the couplings while maintaining a constant ratio of couplings
Ωbc/Ωbd=Pc/Pd. Properties ρ, δ, λopt are the same as Table 1. Distances between |a〉, |c〉, and |d〉 are all 21μm to minimize cross-talk.

Waveguidelet

|a〉

|b〉1

|b〉2

|b〉3

|b〉4

|b〉5

|c〉

|d〉

Lopt(mm)

N/A

21.446

25.440

24.823

22.198

13.554

N/A

N/A

x (μm)

0

13.328

12.188

11.712

11.246

10.246

18.187

18.187

y (μm)

0

0.314

0.329

0.336

0.343

0.361

+10.500

−10.500

Tables (3)

Table 1

Device geometry and parameters used in all calculations regarding the three-state coupler. DAP is from |a〉 to |c〉, and the central waveguide, |b〉, is split into 5 waveguidelets, |b〉1 to |b〉5. Propagation occurs in the z-direction and all segments are aligned at y = 0. |a〉, |b〉1, |c〉 all begin at z = 0. Segment |b〉i+1 is connected to the end of segment |b〉i. The x-coordinate describes the centre of the waveguide. Segments are varied in the x-direction to vary the couplings.

Waveguidelet

|a〉

|b〉1

|b〉2

|b〉3

|b〉4

|b〉5

|c〉

Lopt(mm)

N/A

7.869

11.270

11.804

11.270

7.869

N/A

x (μm)

10.500

−1.177

−0.355

0.000

0.355

1.177

−10.500

ρ

3 μm

δ

0.0045

λopt

800 nm

Table 2

Device geometry and parameters used in transport calculations for five-state based 50:50 splitter. DAP is from |c〉 to |a〉 and |e〉, and the intermediate waveguides, |b〉, |d〉, are split into 5 waveguidelets each, |b〉1, |d〉1 to |b〉5, |d〉5. Propagation occurs in the z-direction and all segments are aligned at y = 0. |a〉, |b〉1, |c〉, |d〉1, |e〉 all begin at z = 0. Segment |b〉i+1 (|d〉i+1) is connected to the end of segment |b〉i (|d〉i). The x-coordinate describes the centre of the waveguide. Segments are varied in the x-direction to vary the couplings. Recall that the device is symmetric about |c〉, and the positions of |a〉, |e〉 and |b〉, |d〉 are related by x → −x. Properties ρ, δ, λopt are the same as Table 1.

Waveguidelet

|a〉

|b〉1

|b〉2

|b〉3

|b〉4

|b〉5

|c〉

Lopt(mm)

N/A

6.606

9.471

9.924

9.480

6.626

N/A

x (μm)

−21.000

−11.959

−11.138

−10.783

−10.427

−9.606

0.000

Table 3

Device geometry and parameters used in transport calculations for four-state based 1/3:2/3 splitter. DAP is from |a〉 to |c〉 and |d〉, and the intermediate waveguide, |b〉, is split into 5 waveguidelets, |b〉1 to |b〉5. Propagation occurs in the z-direction. |a〉, |b〉1, |c〉, |d〉 all begin at z = 0. Segment |b〉i+1 is connected to the end of segment |b〉i. The x and y-coordinates describe the centre of the waveguides. Segments are varied in the x and y-directions to vary the couplings while maintaining a constant ratio of couplings
Ωbc/Ωbd=Pc/Pd. Properties ρ, δ, λopt are the same as Table 1. Distances between |a〉, |c〉, and |d〉 are all 21μm to minimize cross-talk.